Preheat the oven to 350 degrees F (175 degrees C).

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Mix cream cheese with 2 cups salsa and 2 chopped green onions in a bowl until thoroughly combined; lightly stir 2 cups shredded Cheddar and 2 cups Monterey Jack cheese into cream cheese mixture. Spoon a scant 1/2 cup of cheese filling in a line down the center of each tortilla and roll tortilla into a log around the filling. Repeat with remaining tortillas and cheese filling. Place rolled tortillas into a 9×13-inch baking dish.

Spread top of enchiladas with remaining 1 cup salsa. Lightly mix remaining 1/2 cup Cheddar cheese and 1/2 cup Monterey Jack cheese in a bowl and spread over the salsa. Sprinkle with remaining green onions. Cover dish with aluminum foil.

Bake in preheated oven until cheese is melted and bubbly, about 20 minutes.